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This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.

This brief investigates the problem of state estimation of quaternion-valued neural networks (QVNNs) with time-varying delays. First, by extending the Jensen inequality to quaternion domain, an extended Jensen inequality with quaternion term is derived. Second, a class of proportional-integral state estimator (PISE) with exponential decay term is proposed. Then, by constructing a suitable Lyapunov-Krasovskii functional (LKF), some sufficient conditions are obtained to ensure the existence of the designed PISE and the gain matrices of the designed PISE can be directly computed. Simulations are given to illustrate the advantage of the proposed method.

Limit distribution theory in the econometric literature for functional coefficient cointegrating regression is incorrect in important ways, influencing rates of convergence, distributional properties, and practical work. The correct limit theory reveals that components from both bias and variance terms contribute to variability in the asymptotics. The errors in the literature arise because random variability in the bias term has been neglected in earlier research. In stationary regression this random variability is of smaller order and can be ignored in asymptotic analysis but not without consequences for finite sample performance. Implications of the findings for rate efficient estimation are discussed. Simulations in the Online Supplement provide further evidence supporting the new limit theory in nonstationary functional coefficient regressions.

The aim of this work is to study the rigidity problem for Steiner's inequality for the anisotropic perimeter, that is, the situation in which the only extremals of the inequality are vertical translations of the Steiner symmetral that we are considering. Our main contribution consists in giving conditions under which rigidity in the anisotropic setting is equivalent to rigidity in the Euclidean setting. Such conditions are given in term of a restriction to the possible values of the normal vectors to the boundary of the Steiner symmetral (see Corollary 1.17, and Corollary 1.18).

In this paper, we analyze the global asymptotic stability and global exponential stability with respect to the Clifford-valued neutral-type neural network (NN) models with time delays. By considering the neutral term, a Clifford-valued NN model with time delays is formulated, which encompasses real-valued, complex-valued, and quaternion-valued NN models as special cases. In order to achieve our main results, the n -dimensional Clifford-valued NN model is decomposed into 2 m n -dimensional real-valued models. Moreover, a proper function is constructed to handle the neutral term and prove that the equilibrium point exists. Utilizing the homeomorphism theory, linear matrix inequality as well as Lyapunov functional methods, we derive the sufficient conditions corresponding to the existence, uniqueness, and global asymptotic stability with respect to the equilibrium point of the Clifford-valued neutral-type NN model. Numerical examples to demonstrate the effectiveness of the results are provided, and the simulations results are analyzed and discussed.

A biased assimilation model of opinion dynamics is a nonlinear model, in which opinions exchanged in a social network are multiplied by a state-dependent term having the bias as exponent and expressing the bias of the agents toward their own opinions. The aim of this article is to extend the bias assimilation model to signed social networks. We show that while for structurally balanced networks, polarization to an extreme value of the opinion domain (the unit hypercube) always occurs regardless of the value of the bias, for structurally unbalanced networks, a stable state of indecision (corresponding to the centroid of the opinion domain) also appears, at least for small values of the bias. When the bias grows and passes a critical threshold, which depends on the amount of "disorder" encoded in the signed graph, then a bifurcation occurs and opinions become again polarized. Funding Agencies|National Natural Science Foundation of China [61733018]; Australian Research Council [DP190103615]; Swedish Research Council [2020-03701]

We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$., Comment: 23 pages

In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.

In this study, the generalized modified variable-coefficient KdV equation with external-force term (gvcmKdV) describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained. Moreover, several new solutions have been constructed for the gvcmKdV. Additionally, the classical direct similarity reduction method is used to reduce the gvcmKdV to a nonlinear ordinary differential equation. Building on the solutions given in the previous literature for the reduced equation, many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.

We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in L 2 and a right-hand side which is only in L 1 . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45(7) (2020) 690–713). Here, we prove its stability by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system.

Addition-min fuzzy relation inequalities are a recently proposed model for describing the transmission mechanism in a peer-to-peer (P2P) network system. Considering the random line faults that take place between two terminals in a P2P network system, we propose a new kind of model, namely, random-term-absent (RTA) addition-min fuzzy relation inequalities. Some of their relevant properties are investigated in this paper. In addition, to decrease the network congestion under a fixed priority grade of the terminals, we introduce the lexicographic minimum solutions of RTA addition-min fuzzy relation inequalities. An efficient algorithm is developed to find the unique lexicographic minimum solution with an illustrative example.

An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper.

We obtain an integral representation for certain functionals arising in the context of optimal design and damage evolution problems under non-standard growth conditions and perimeter penalisation. Under our hypotheses, the integral representation includes a term which is absolutely continuous with respect to the Lebesgue measure and a perimeter term, but no additional singular term. We also study some dimension reduction problems providing results for the optimal design of thin films., Comment: arXiv admin note: text overlap with arXiv:1712.03657

We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy., Comment: 39 pages, final version, IMRN

We present a gentle approach to the justification of effective media approximations, for PDE's set outside the union of $n \gg 1$ spheres with low volume fraction. To illustrate our approach, we consider three classical examples: the derivation of the so-called strange term, made popular by Cioranescu and Murat, the derivation of the Brinkman term in the Stokes equation, and a scalar analogue of the effective viscosity problem. Under some separation assumption on the spheres, valid for periodic and random distributions of the centers, we recover effective models as $n$ goes to infinity by simple arguments., 20 pages. Final version, to appear in Asympt. Anal

This study focuses on the stability analysis of sampled-data systems. The idea is to propose a novel Lyapunov functional method using a new framework. In the functional method, the new discontinuous term satisfying V 1 ( t k - ) ⩾ 0 and V 1 ( t k ) = 0 or V 2 ( t k - ) = 0 and V 2 ( t k ) ≤ 0 is different from and further extends existing methods; the common term V 0 ( t ) = x T ( t ) Px ( t ) has a more relaxed condition than the positive definiteness generally required in the literature. Based on this method, a series of stability criteria in terms of linear matrix inequalities (LMIs) are obtained and demonstrated to be theoretically less conservative than some recent results. Finally, two numerical examples are provided to show the effectiveness and advantages of our results compared with those of previous studies.

In this paper, linearly implicit backward differentiation formulas with variable step-sizes are proposed to solve numerically the two-dimensional incompressible Navier-Stokes equations (formulated in vorticity-stream function). With Fourier pseudospectral methods for spatial discretization, the diffusion term is discretized implicitly and the nonlinear convection term is treated by a combination of implicit and explicit discretizations. As a result, only linear solvers are needed at each time step to achieve the desired temporal accuracy. With the help of a priori assumption and aliasing error control techniques, the error estimates for one-step and two-step backward differentiation formulas are established in several norms under appropriate step-size constraints. Compared with the numerical results of implicit-explicit (the nonlinear convection term is treated explicitly) BDF2 method and fully implicit Crank-Nicolson method, it demonstrates that the proposed linearly implicit variable step-size BDF2 method is effective and robust.

In the present endeavour, the shifted Legendre collocation method is extended to obtain the solution of nonlinear fractional order (2+1)-dimensional advection–reaction–diffusion solute transport equation. The variations of solute concentration of the model for different fractional order space and time derivatives are presented graphically for various particular cases. The main feature of the present contribution is the graphical exhibitions of the effects of advection term, reaction term and fractional-order parameters on the solution profile. To authenticate the effectiveness of the method, a drive has been taken to compare the obtained results with the existing analytical results of the integer-order form of the considered model through error analysis which are displayed in tabular and pictorial forms.

In the current work we have used the deep learning based long short-term memory model to predict the strength and peak time of solar cycle 25 by employing the monthly smoothed sunspot number data obtained from WDC-SILSO, Royal Observatory of Belgium, Brussels. We have used the stacked LSTM forecasting model to predict the upcoming cycle 25. From our analysis it has been shown that our proposed model is capable of capturing long term dependencies as well as trend within the data. For cycle 20 and 21 the error difference between predicted as well as observed peak value is 2.3 and 0.7 respectively while the peak prediction error is 1.47% and 0.30%. The RMSE of the model for cycle 20 and 21 is 3.97 and 4.34 respectively. For cycle 22, the AE and RE is 4.6 and 2.16% while the RMSE of the model for this case is 4.50. The predicted peak amplitude of solar cycle 23 and 24 from our proposed model has a relative error of 1.75% and 1.99% respectively from the observed value while the RMSE is 3.4 for cycle 23 and 4.2 for cycle 24. Our projected prediction of cycle 25 using the proposed LSTM model, says that it will be stronger than cycle 24 and weaker than cycle 23. The solar cycle 25 will peak with an amplitude of sunspot number at 171.9 ± 3.4 and will be 47 % stronger than cycle 24. The solar cycle 25 will reach its peak in August 2023 ± 2 months.

We study the asymptotic behavior of the Kuramoto system with time-periodic natural frequencies and all-to-all coupling. For a relatively large coupling strength K > 0 compared to the differences between the time-periodic natural frequencies, we obtain the existence of time-periodic states variables θ i j = θ i − θ j . We derive an explicit formula for the total phase of the solution, which depends on the initial data and natural frequencies. We prove that if the solutions to the Kuramoto system with all-to-all coupling have the same total phase, then their phase differences θ i j for any corresponding initial data converge to the same periodic state θ i j T as t goes to ∞. Finally, by combining the total phase formula and this stability, we present a sharp asymptotic formula consisting of the time-periodic state, drift term, and exponentially decaying error term. The drift term depends only on the natural frequency and initial data. Several simulations are carried out to validate the proposed results.

Accurate quantification of photovoltaic (PV) system degradation rate (RD) is essential for lifetime yield predictions. Although RD is a critical parameter, its estimation lacks a standardized methodology that can be applied on outdoor field data. The purpose of this paper is to investigate the impact of time period duration and missing data on RD by analyzing the performance of different techniques applied to synthetic PV system data at different linear RD patterns and known noise conditions. The analysis includes the application of different techniques to a 10-year synthetic dataset of a crystalline Silicon PV system, with emulated degradation levels and imputed missing data. The analysis demonstrated that the accuracy of ordinary least squares (OLS), year-on-year (YOY), autoregressive integrated moving average (ARIMA) and robust principal component analysis (RPCA) techniques is affected by the evaluation duration with all techniques converging to lower RD deviations over the 10-year evaluation, apart from RPCA at high degradation levels. Moreover, the estimated RD is strongly affected by the amount of missing data. Filtering out the corrupted data yielded more accurate RD results for all techniques. It is proven that the application of a change-point detection stage is necessary and guidelines for accurate RD estimation are provided.

This letter proposed a spatial and spectral anisotropic tensor total variation (SSATTV) driven adaptive pansharpening model for the fusion of low-resolution (LR) multispectral (MS) and panchromatic (Pan) images to the high-resolution (HR) MS images. Except for the local spectral consistency constraint-based fidelity term between HR and LR MS used for spectral preservation, the proposed model reformulated the adaptive linear constraint-based fidelity term between Pan and HR MS with the tensor-representation modeling, and particularly proposed a SSATTV prior term which imposed the spatial anisotropic TV prior between HR MS and Pan for spatial gradient feature preservation, and the spectral TV sparsity prior on HR MS for further spectral feature preservation. Furthermore, the proposed model was efficiently solved via the alternating direction method of multipliers (ADMM) scheme. Specifically, the experiments validated the superiority of proposed SSATTV method.

This paper focuses on the global Mittag–Leffler (M–L) synchronization for fractional-order inertial neural networks (FOINNs) including the Caputo derivative and delay. By choosing an appropriate variable transformation, the considered system including the inertial term is transformed into two fractional-order nonlinearly coupled interconnected subsystems. Combining with the delayed-feedback controller, fractional Lyapunov functional approach and inequality analysis technique, multi sets of novel algebraic criteria on the global M–L synchronization between derive-response systems are established. The main prominent highlight of this paper is that the proposed results are characterized by the algebraic inequalities form, which can simplify the calculation and facilitate the test. Two numerical simulation examples validate the theoretical results.

This study investigates the fundamental limits of variable-length compression in which prefix-free constraints are not imposed (i.e., one-to-one codes are studied) and non-vanishing error probabilities are permitted. Due in part to a crucial relation between the variable-length and fixed-length compression problems, our analysis requires a careful and refined analysis of the fundamental limits of fixed-length compression in the setting where the error probabilities are allowed to approach either zero or one polynomially in the blocklength. To obtain the refinements, we employ tools from moderate deviations and strong large deviations. Finally, we provide the third-order asymptotics for the problem of variable-length compression with non-vanishing error probabilities. We show that unlike several other information-theoretic problems in which the third-order asymptotics are known, for the problem of interest here, the third-order term depends on the permissible error probability., Comment: 14 pages, Accepted by the IEEE Transactions on Information Theory

A second order multirate scheme for the evolutionary coupled Stokes–Darcy problem is proposed and analyzed. The scheme decouples the problem into two smaller subproblems at each time step by exploiting the mismatch in flow activity between the domains. Specifically, the slower Darcy subproblem is integrated with a larger time step compared to the more active Stokes subproblem. This avoids expending unnecessary computational effort on sampling the slow changing Darcy solution. The slow varying Darcy solution is extrapolated/interpolated onto the fine grid for the integration of the Stokes subproblem. The long-time stability and convergence of the method is proved under a time step restriction that depends on physical parameters and the ratio of the time steps in each subdomain. The scheme includes a stabilization term that relaxes the time step restriction. Numerical experiments confirm convergence of the method and the effectiveness of the stabilization technique at relaxing the stability time step restriction.

This paper presents a new asset pricing model incorporating fundamental uncertainties by choice of a probability measure. This approach is novel in that we incorporate uncertainties on Brownian motions describing risks into the existing asset pricing model. Particularly, we show extensions of interest rate models to the ones with uncertainties on the Brownian motions, which make the yield curve reflect not only economic factors but also views of the market participants on the Brownian motions. Such yield curve models are especially important in yield curve trading of hedge funds as well as monetary policy making of central banks under low interest rate environments observed after the global financial crisis, in which yield curves are less affected by economic factors since they are controlled by the central banks, but are driven mainly by sentiments of market participants. Firstly, to model aggressive (positive)/conservative (cautious) attitudes towards such fundamental uncertainties, we consider a sup-inf/inf-sup problem on the utility of a representative agent with respect to uncertainties over Brownian motions, i.e. fundamental market risks, by choice of a probability measure. Secondly, we show that the problem is solved via a backward-stochastic differential equations (BSDEs) approach. Then, under a probability measure determined by solving the sup-inf/inf-sup problem, we propose interest rate models with those uncertainties and explicitly obtain their term structures of interest rates. Particularly, we present two approaches to solving the relevant coupled forward-backward stochastic differential equations (FBSDEs) to obtain expressions of the equilibrium interest rate and the term structure of interest rates. In detail, the first approach is by comparison theorems, and the second approach is to predetermine the signs of the volatilities of the BSDE in the coupled system and confirm them by explicitly solving the separated BSDE. Finally, we present concrete examples with numerical experiments.

Two experiments were conducted to determine standardized ileal digestibility (SID) of amino acids (AA) and the concentration of metabolizable energy (ME) in non-heat-treated and heat-treated soybean expellers (L-0, L-12, and L-48). L-0 underwent short-term steam conditioning for 60 s, whereas L-12 and L-48 underwent short-term steam conditioning for 60 s and long-term steam conditioning for 12 or 48 min. All heat-treated soybean expellers were expander processed. In experiment 1, 10 ileal-cannulated barrows (54.22 ± 4.54 kg) were allotted to a replicated 5 × 4 Youden square design with eight replicate pigs per diet. Each source of soybean expellers was included in one diet, and a nitrogen-free diet was also used. Results indicated that the SID of AA in non-heat-treated soybean expellers was less (P

In this article, we study a hyperbolic equation of Cahn–Hilliard with a proliferation term and Dirichlet boundary conditions. In particular, we prove the existence and uniqueness of the solution, and also the existence of the global attractor.

A new estimator is proposed in this paper that inverts a binding function that is based on the asymptotic bias of the within-groups (WG) estimator in a first-order dynamic panel model with fixed effects under the asymptotic regime of large N and finite T. The new estimator is in the spirit of indirect inference (II) and corrects the inconsistency of the WG estimator. It is simulation-free, does not rely on any distributional assumption on the idiosyncratic error term, and is asymptotically normally distributed. Monte Carlo results indicate its good finite-sample performance in comparison with other consistent estimators.

In this paper an exact rejection algorithm for simulating paths of the coupled Wright–Fisher diffusion is introduced. The coupled Wright–Fisher diffusion is a family of multivariate Wright–Fisher diffusions that have drifts depending on each other through a coupling term and that find applications in the study of networks of interacting genes. The proposed rejection algorithm uses independent neutral Wright–Fisher diffusions as candidate proposals, which are only needed at a finite number of points. Once a candidate is accepted, the remainder of the path can be recovered by sampling from neutral multivariate Wright–Fisher bridges, for which an exact sampling strategy is also provided. Finally, the algorithm’s complexity is derived and its performance demonstrated in a simulation study.

In the Cauchy problem L1u≡Lu+(b,∇u)+cu-ut=0,(x,t)∈D,u(x,0)=u0(x),x∈RN, for nondivergent parabolic equation with growing lower-order term in the half-space D=RN×[0,∞), N⩾3, we prove sufficient conditions for exponential stabilization rate of solution as t→+∞ uniformly with respect to x on any compact K in RN with any bounded and continuous in RN initial function u0(x).

This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The frac...

This study aims to analyze the short-term and long-term relationship of increasing the minimum wage in Central Java on employment. The research method used is ECM. The variables of this study include labor, minimum wages, PMDN, and economic growth. The data used are time-series data from 1990-2020. The results show that the minimum wage has a positive and significant relationship to the employment in the long term but not significantly in the short time. PMDN has a negative but significant correlation in the short and long term. At the same time, the variable economic growth has a positive but not meaningful relationship to employment absorption in the long and short term.

This work is concerned with the study of generalized travelling wave solutions for time heterogeneous Fisher-KPP equations with nonlocal diffusion. Here we consider general time heterogeneities both for the diffusion kernel and the reaction term. We investigate the existence and non existence of generalized travelling wave solutions for such a problem. Roughly speaking we prove that generalized travelling waves do exist for all sufficiently large wave speed function in some average sense, while such solutions do not exist for speed function with small average. In addition, under suitable assumptions on the time varying coefficients, we derive a sharp estimate for the average speed functions of the generalized travelling wave solutions.

We provide a Serrin type blow-up criterion for the 3-D viscous compressible flows with large external potential force. For the Cauchy problem of the 3-D compressible Navier-Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup-norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.

The present work extends recent results by second author concerning sampled-data feedback stabilization for affine in the control of nonlinear systems with nonzero drift term, under the presence of a generalized control Lyapunov function associated with appropriate Lie algebraic hypotheses concerning the dynamics of the system. The main results of present work, constitute a generalization of the well-known "Artstein-Sontag" theorem on asymptotic stabilization by means of an almost smooth feedback controller. The analysis is limited to the affine single-input nonlinear systems with nonzero drift term, however, the results can easily be extended to the multi-input case. An illustrative example of the derived results is included., Comment: 7 pages, submitted to IEEE Transactions on Automatic Control for possible publication

In the present paper we study the existence of ground state solutions for Henon equations involving with single or multiple upper weighted critical exponents with general perturbational term, where the upper weighted critical exponents include upper Hardy-Sobolev, Sobolev or Henon-Sobolev critical exponents for the embedding from H r 1 ( R N ) into L q ( R N ; | x | α ) with α > − 2 . The Nehari manifold method and the mountain pass theorem are applied to obtain the ground state solutions with different assumptions on general perturbational term. The existence of ground state solutions is closely related to the sign of the parameters in the equations.

In this paper, we study the mean square exponential stability of discrete-time stochastic neural networks with partially unstable subsystems and mixed delays. The mixed delays under consideration involve discrete delay and distributed delay. Moreover, the discrete delay term satisfies the Bernoulli distribution . Different from the deterministic switching, we consider Markov switching and our system has partially unstable subsystems. By constructing a novel Lyapunov–Krasovskii functional and using the stationary distribution of Markov chain , we give sufficient conditions for the mean square exponential stability of the suggested system. Finally, two numerical examples are given to check the theory results.

Significant wave height prediction for the following hours is a necessity for the planning and operation of wave energy devices. For a site-specific and short-term prediction, classical numerical wave forecasting methods may not be justified as exhaustive climatological data and huge computational power are needed. In this paper, a combination of a decomposition approach and long short-term memory network was presented to forecast the significant wave heights. An improved version of complete ensemble empirical mode decomposition algorithm and recurrence quantification analysis were applied to separate the original time series into deterministic and stochastic components. Each decomposed series was forecasted by the long short-term memory network and the final predicted significant wave heights were obtained by integrating the deterministic and stochastic predictions. Wave data measured at three buoy stations along the eastern coast of the United States were utilized to verify the hybrid model. The performance of the proposed method in three different wave height ranges was evaluated. The results suggested that the hybrid model outperformed the stand-alone long short-term memory network adjusted on the unseparated signal; in particular, for longer lead times and larger wave heights.

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance $\varepsilon$ is at most $O(2^{0.2680n})$ times a polynomial in $1/\varepsilon$. On bipartite graphs, the exponential term in the running time is improved to $O(2^{0.2372n})$. Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, \v{S}tefankovi\v{c} and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless $\mbox{P}=\mbox{NP}$., Comment: 52pp

Consider the heterogeneity (e.g., heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different numbers of sexual partners etc.) of host population, in this paper, the authors propose an infection age multi-group SEIR epidemic model. The model system also incorporates the feedback variables, where the infectivity of infected individuals may depend on the infection age. In the direction of mathematical analysis of model, the basic reproduction number R0 has been computed. The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R0. More precisely, for R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and for R0 > 1, they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method. By considering a numerical example, they investigate the effects of infection age and feedback on the prevalence of the disease. Their result shows that feedback parameters have different and even opposite effects on different groups. However, by choosing an appropriate value of feedback parameters, the disease could be eradicated or maintained at endemic level. Besides, the infection age of infected individuals may also change the behaviour of the disease, global stable to damped oscillations or damped oscillations to global stable.

In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Holder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger than one. In the latter case, the estimates are robust enough to conclude interior C 1 , α regularity by an improvement of the flatness procedure, which is possible when the nonlocal term is close enough to a second-order diffusion.

Long-term prediction parameters of polar motion (PM) with high precision have an important significance in understanding Earth geophysical processes in the future. The multilayer perceptron (MLP) is introduced into PM prediction and we try for the first time to combine MLP with autoregressive moving average model (ARMA) and singular spectrum analysis (SSA) for the mid-long-term prediction of PM. The principal components of PM were extracted and predicted by SSA first. Then, the combined MLP and ARMA were used to predict the residual term. The PM predictions were obtained by adding the principal components and the residual predictions. Applying this proposed method, 157 500-day and 12 5-year lead time predictions of PM were made based on IERS 14 C04 product. The results showed that the proposed method performed well in the mid-long-term period of PM predictions, which was superior to IERS Bulletin A.

In this paper, we consider a one-dimensional porous system damped with a single weakly nonlinear feedback and distributed delay term. Without imposing any restrictive growth assumption near the origin on the damping term, we establish an explicit and general decay rate, using a multiplier method and some properties of convex functions in case of the same speed of propagation in the two equations of the system. The result is new and opens more research areas into porous-elastic system.

A decision list is an ordered list of rules. Each rule is specified by a term, which is a conjunction of literals, and a value. Given an input, the output of a decision list is the value corresponding to the first rule whose term is satisfied by the input. Decision lists generalize both CNFs and DNFs, and have been studied both in complexity theory and in learning theory. The size of a decision list is the number of rules, and its width is the maximal number of variables in a term. We prove that decision lists of small width can always be approximated by decision lists of small size, where we obtain sharp bounds. This in particular resolves a conjecture of Gopalan, Meka and Reingold (Computational Complexity, 2013) on DNF sparsification. An ingredient in our proof is a new random restriction lemma, which allows to analyze how DNFs (and more generally, decision lists) simplify if a small fraction of the variables are fixed. This is in contrast to the more commonly used switching lemma, which requires most of the variables to be fixed., Comment: 16 pages